52 AOE 5104 Class 4 9408

AOE 5104 Class 9/4/08 • Online presentations for today’s class: – Vector Algebra and Calculus and • • • • Vector Algebra and Calculus Crib Homework Homework due 9/11 Study group assignments have been made and are online • Recitations will be – Mondays @ 5:30pm (with Nathan Alexander) in Randolph 221 – Tuesdays @ 5pm (with Chris Rock) in Whitemore 349 I have added the slides without numbers The numbered slides are the original file Last Class ez • Changes in Unit Vectors • Calculus w.r.t time • Integral calculus w.r.t space • Today: differential calculus in 3D P' P e r z eθ r θ deθ = −dθe r de r = dθeθ de z = dθ ∂ ( A + B ) ∂A ∂B = + ∂t ∂t ∂t ∂ ( A.B ) ∂B ∂A =A + B ∂t ∂t ∂t ∂( A × B ) ∂B ∂A = A× + ×B ∂t ∂t ∂t ∫ ( A + B ) dt = ∫ Adt + ∫ Bdt Oliver Heaviside 1850-1925 Shock in a CD Nozzle Bourgoing & Benay (2005), ONERA, France Schlieren visualization Sensitive to in-plane index of ref gradient Differential Calculus w.r.t Space Definitions of div, grad and curl In 1-D df   = lim  [ f ( x + ∆x) − f ( x) ] ÷ dx ∆x →  ∆x  In 3-D   gradφ ≡ lim  Ñ φ ndS ÷ δ τ→0 δ τ ∫  ΔS    divD ≡ lim  Ñ D.ndS ÷ ∫ δ τ →0 δ τ  ΔS    curlD ≡ − lim  Ñ D × ndS ÷ δ τ →0 δ τ ∫  ΔS  D=D(r), φ = φ (r) n dS Elemental volume δτ with surface ∆S Gradient φ= h hig φ ndS (medium) φ= n φ ndS (large) low Resulting φ ndS dS φ ndS (small) φ ndS (medium)   gradφ ≡ lim  Ñ φ ndS ÷ ∫ δ τ →0 δ τ  ΔS  Elemental volume δτ with surface ∆S = magnitude and direction of the slope in the scalar field at a point Review Gradient gradφ ≡ Limδτ →0 φndS ∫ δτ ΔS Magnitude and direction of the slope in the scalar field at a point Gradient low • Fourier´s Law of Heat Conduction ∂T q = − k = − k∇T n ∂n h hig φ= ∂φ e s ∇φ = ∂s s, e s φ= • Component of gradient is the partial derivative in the direction of that component ∇φ Differential form of the Gradient Cartesian system Evaluate integral by expanding the variation in φ about a point P at the center of an elemental Cartesian volume Consider the two x faces: ∂φ dx   φ n dS ≈ φ −  (−i )dydz ∫ ∂x   Face ∂φ dx   φ n dS ≈ φ +  (+i )dydz ∫ ∂x   Face ∂φ i dxdydz adding these gives ∂x gradφ ≡ Limδτ→0 φndS ∫ δτ ΔS φ = φ(x,y,z) k P dz i j Face Proceeding in the same way for y and z we get j ∂φ ∂φ dxdydz, so dxdydz and k ∂z ∂y Face dx dy φndS ∫ δτ ΔS  ∂φ  ∂φ ∂φ ∂φ ∂φ ∂φ  = Limδτ→0  i dxdydz + j dxdydz + k dxdydz  = i + j +k δτ  ∂x ∂y ∂z ∂x ∂y ∂z  gradφ ≡ Limδτ→0 1st Order Integral Theorems • ∫ ∇φdτ = ∫ φndS Gradient theorem R • Curl theorem Stokes’ theorem dτ S ∫ ∇ × Adτ = −∫ A × ndS R • S ∫ ∇.Adτ = ∫ A.ndS Divergence theorem R • ndS S ∫ ∇ × A.nd S = ∫ A.ds S Volume R with Surface S C Open Surface S with Perimeter C ndS The Gradient Theorem Finite Volume R Surface S Begin with the definition of grad: gradφ ≡ Limδτ→0 φndS ∫ δτ ΔS Sum over all the dτ in R: ∑ gradφ dτ = ∑ ∫ φndS dτ i R R ΔSi We note that contributions to the RHS from internal surfaces between elements cancel, and so: ∑ gradφ dτ = ∑ φndS R dτi+1 S Recognizing that the summations are actually infinite: ∫ gradφ dτ = ∫ φndS R nidS S ni+1dS dτi Assumptions in Gradient Theorem ∫ gradφ dτ = ∫ φndS R S • A pure math result, applies to all flows • However, S must be chosen so that φ is defined throughout R ∫ gradφ dτ = ∫ φndS R S +Σ Submarine surface Σ S Flow over a finite wing S1 S1 S2 S = S1 + S2 R is the volume of fluid enclosed between S1 and S2 − ∫ ∇p dτ = − ∫ pndS R S p is not defined inside the wing so the wing itself must be excluded from the integral 1st Order Integral Theorems • ∫ ∇φdτ = ∫ φndS Gradient theorem R • Curl theorem Stokes’ theorem dτ S ∫ ∇ × Adτ = −∫ A × ndS R • S ∫ ∇.Adτ = ∫ A.ndS Divergence theorem R • ndS S ∫ ∇ × A.nd S = ∫ A.ds S Volume R with Surface S C Open Surface S with Perimeter C ndS Alternative Definition of the Curl e Perimeter Ce Area δσ ds e.curlA ≡ Limδσ →0 δσ ∫ A.ds = Limδσ Ce →0 ΓCe δσ Stokes’ Theorem Begin with the alternative definition of curl, choosing the direction e to be the outward normal to the surface n: n.∇ × A ≡ Limδσ →0 δσ ∫ A.ds Ce Finite Surface S With Perimeter C n Sum over all the dσ in S: ∑ n.∇ × Adσ i = ∑ ∫ A.ds S dσ S Ce Note that contributions to the RHS from internal boundaries between elements cancel, and so: ∑ n.∇ × Adσ = ∑ A.ds S C Since the summations are actually infinite, and replacing σ with the more normal area symbol S: ∫ ∇ × A.nd S = ∫ A.ds S C dsi+1 dsi dσi+1 dσi Stokes´ Theorem and Velocity ∫ ∇ × A.nd S = ∫ A.ds S C • Apply Stokes´ Theorem to a velocity field ∫ ∇ × V.nd S = ∫ V.ds S C • Or, in terms of vorticity and circulation ∫ Ω.nd S = ∫ V.ds = Γ C S C • What about a closed surface? ∫ Ω.ndS = S Assumptions of Stokes´ Theorem ∫ ∇ × A.nd S = ∫ A.ds S C • A pure math result, applies to all flows • However, C must be chosen so that A is defined over all S ∫ Ω.nd S = ∫ V.ds ? S C C The vorticity doesn’t imply anything about the circulation around C 2D flow over airfoil with Ω=0 Flow over a finite wing ∫ ∇ × V.nd S = ∫ V.ds S C C S Wing with circulation must trail vorticity Always Vector Operators of Vector Products ∇(ψΦ )  ∇.( ΦA)  ∇ × ( ΦA)  ∇(A.B)   ∇.(A × B)   ∇ × (A × B) = ψ ∇Φ + Φ ∇ψ   = Φ∇.A + ∇Φ A   = Φ ∇ × A + ∇Φ × A        = (A.∇ )B + (B.∇A) + A × ( ∇ × B) + B × ( ∇ × A)     = B.∇ × A - A.∇ × B         = A( ∇.B) + (B.∇ )A - B( ∇.A) - (A.∇ )B Convective Operator  (A.∇ )Φ  ∂ ∂ ∂ =  Ax + Ay + Az φ ∂y ∂z   ∂x  = A.( ∇Φ )   (A.∇ )B  ∂ ∂ ∂  =  Ax + Ay + Az  B ∂z   ∂x  ∂y    = 12 [ ∇(A.B) - A × ( ∇ × B) - B × ( ∇ × A)      - ∇ × (A × B) + A( ∇.B) - B( ∇.A)] V.∇ρ = change in density in direction of V, multiplied by magnitude of V Second Order Operators ∂ 2φ ∂ 2φ ∂ 2φ ∇.∇φ = ∇ φ = + + ∂x ∂y ∂z The Laplacian, may also be applied to a vector field ∇(∇.A) ∇ × ∇ × A = ∇(∇.A) − ∇ A ∇ × ∇φ ≡ • So, any vector differential equation of the form ∇ × B=0 can be solved identically by writing B=∇ φ • We say B is irrotational • We refer to φ as the scalar potential ∇.∇ × A ≡ • So, any vector differential equation of the form ∇ B=0 can be solved identically by writing B=∇ × A • We say B is solenoidal or incompressible • We refer to A as the vector potential Class Exercise Make up the most complex irrotational 3D velocity field you can V = (e sin x cos x + xy )i + x y j − k / z ? We can generate an irrotational field by taking the gradient of any scalar field, since ∇ × ∇φ ≡ I got this one by randomly choosing φ = esin x + x y + / z And computing ∂φ ∂φ ∂φ V= i+ j+ k ∂x ∂y ∂z 2nd Order Integral Theorems • Green’s theorem (1st form) ψ ∇ ∫ φ + ∇ ψ ∇ φ d τ = ∫ψ R S ∂φ dS ∂n Volume R with Surface S ndS • Green’s theorem (2nd form) dτ  ∂φ ∂ψ  ψ ∇ φ − φ ∇ ψ d τ = ( ) ∫ Ñ ∫ ψ ∂n -φ ∂n ÷ dS R S These are both re-expressions of the divergence theorem ...I have added the slides without numbers The numbered slides are the original file Last Class ez • Changes in Unit Vectors • Calculus w.r.t time • Integral calculus w.r.t space • Today: